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Optical depth profiling by attenuated total reflection Fourier transform infrared spectroscopy using an incident beam with arbitrary degree of polarization
VIBRATIONAL SPECTROSCOPY ELSEVIER
Vibrational Spectroscopy 13 (1996) 1-9
Optical depth profiling by attenuated total reflection Fourier transform infrared spectroscopy using an incident beam with arbitrary degree of polarization Sanong Ekgasit, Hatsuo Ishida * Department of Macromolecular Science, Case Western Reserve University, Cleveland, OH 44106-7202. USA
Received 16 January 1996; accepted 6 April 1996
Abstract A depth profiling analysis via multiple angle attenuated total reflection Fourier transform infrared spectroscopy using an incident beam with arbitrary degree of polarization has been developed. A guessed complex refractive index profile is not required for the analysis. The analysis consists of three steps: estimation of the complex refractive index profile by the estimated mean square electric field, linear least square fitting of absorptances, and nonlinear fitting of reflectances. The applicability of this approach with various types of profile and degrees of polarization has been shown by simulated spectra with added white noise. The effect of amplitudes of noise on the accuracy of this approach has been investigated. Keywords." ATR FTIR; Multiple angle ATR FrlR spectroscopy;Depth profiling; Linear least square fitting; Nonlinear fitting
1. Introduction One of the unique characteristics of ATR FFIR spectroscopy is that it provides depth dependent information as a function of refractive index of the incident medium, complex refractive index of the film, angle, and frequency of the incident beam [1,2]. The depth dependent properties such as concentration, conformation, and degree of polymerization of a film can, then, be obtained from multiple angle ATR spectra. There are two depth profiling techniques that take advantage of this unique characteristic, namely Laplace transform and exact optical theory techniques [3-8]. Although the applicability of
* Corresponding author. Tel.: + 1-216-3684285; fax: + 1-2163684164.
the latter method is not limited to spectral regions with small absorption as is the former method, both methods require an estimated or guessed complex refractive index profile for their calculations. By using the estimated profile, the experimental data is then fitted to the calculated data. Even though the calculations of both techniques are straightforward, it is difficult for one to estimate the right profile even if some information about the profile is known. Recently, we have developed a new depth profiling method that overcomes the above difficulties [ 1]. This method utilizes multiple angle ATR spectra taken with a p or s polarized incident beam. The analysis consists of two steps: linear least square fitting of the absorptances and nonlinear fitting of the reflectances. The estimated complex refractive index profile is obtained by solving a set of linear
0924-2031/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PIIS0924-2031(96)00028-8
2
S. Ekgasit, H. lshida/Vibrational Spectroscopy 13 (1996) 1-9
equations of absorptance. The profile is then used as a trial profile for nonlinear fitting of reflectances from experiments to those from exact optical theory. The purpose of this paper is to develop a multiple angle ATR depth profiling method for a set of spectra taken with an arbitrarily polarized incident beam. This will allow an infrared beam of a spectrometer without a polarizer to be used as long as the degree of polarization caused by the instrument is known. In order to obtain the complex refractive index profile of the film, the same approach for depth profiling calculation mentioned above with some modifications is utilized.
2. Theory
2.1. Spectral intensity with an arbitrary, polarized incident beam A film with anisotropic properties in the depth direction can be considered as a stratified medium. The stratified medium is defined as a stack of n isotropic thin films. Each film or layer of the stratified medium is considered homogeneous and parallel plane-bound with the neighboring layers. The characteristics of the j-th layer is given by its complex refractive index, ~j = nj + ikj, and thickness, dj. The reflectance, transmittance, and absorptance of the film are the same as those of the stratified medium [9,10]. For an ATR spectrum of a thick film, reflection lost from the spectrum is caused solely by the absorption of the film. The relationship between reflectance, Rt(0), and absorptance, At(0), in ATR FTIR spectroscopy is given by [1,4-6,11]:
A,(O) = 1 - R , ( 0 )
(1)
where l indicates polarization (i.e., s or p) and 0 is the incident angle. The absorptance of the film can be expressed in terms of characteristic parameters of the stratified medium as [1-3,11,12]: j ~ 1njkj f.;'+'(E2zt(O))dz At(O ) - - 4"n'v n o cos 0 .=
(2)
where (Ez2/(0)) is the mean square electric field at depth z, zj is the depth at the interface between the
( j - 1)-th and j-th layers, z,+ 1 is the depth where the mean square electric field is negligibly small. By adopting the working depth concept, molecular characteristics beyond the working depth have insignificant contributions to the spectral intensity [1]. The working depth is defined as the depth at which the evanescent field has decayed to 0.25% of its value the surface. The working depth is approximately three times the penetration depth [12]. The depth Zn+ 1 is then defined by the working depth. It should be noted that the evanescent field is the special name for the mean square electric field in a non-absorbing medium (i.e., k = 0). It can be calculated if the refractive index of the film is known. For an arbitrary polarized incident beam, the degree of polarization of the beam, ,~, is given by [13,14]:
ip-l ip+(,
(3)
where lp and I s are p and s polarized intensities of the incident beam, respectively. The degree of polarization lies in the range - 1 < ~ < 1. The degree of polarization of the p polarized incident beam is equal to + 1 while that of the s polarized beam is equal to - 1 . Natural light is defined as an non-polarized beam since the intensities of p and s polarized components of the beam are the same. As a result, the degree of polarization of natural light is equal to zero. Unlike natural light, the beam of a spectrometer is partially polarized [ 15]. Its degree of polarization must be determined in order to use this method for depth profiling calculation. With an incident beam of degree of polarization ~, the spectral intensity of a film is the combination of the intensities corresponding to the p and s components. In ATR FTIR spectroscopy, reflectance and absorptance of a film corresponding to an incident beam with a degree of polarization ~ (i.e., Rp(0) and A~(0)) are given by the following expressions [15,16]:
R~(O) = ½{(1 + f i ) R p ( 0 ) + (1 - , ~ ) R s ( 0 ) } ,
(4)
A~(O) = 2 { ( l + f i ) a p ( 0 )
(5)
+(1-~)as(0)}
where Rp(e) and Ap(0) are reflectance and absorp-
S. Ekgasit, H. lshida / Vibrational Spectroscopy 13 ¢1996) 1 - 9
tance of the film with a p polarized incident beam while R~(0) and A,(0) are those with an s polarized incident beam, respectively.
are obtained, respectively, for p and s polarized incident beams: Ap(0)= 1-Rp(0) 2~-v{1 + Rp(0)}
2.2. The estimated mean square electric field = The mean square electric field is significantly different from the evanescent field, especially at positions where the extinction coefficient is large [1]. For a peak position with a small extinction coefficient, the attenuation effect by the absorption is small. The mean square electric field can be estimated from the reflectance and evanescent field at the same conditions (i.e., refractive indices of incident medium and film, angle, frequency, and polarization of the incident beam). Under the above condition, the estimated mean square electric field is assumed to have the same decay characteristic as that of the evanescent field and is given by [17]: 1 +R,(O)
___-
2
1 + R,(0) 2
3
/ 2 ,E.(o)L=o
n0cos 0
X L n j k j fr z j + 1e- 2 :/drdz, .j= I .# A~(O) = 1 -R~(O) 27rv{1 + Rs(0)}
=
n o cos 0
X ~ n j k j f zJ+l e --:/d~ "~ dz. j=
1
The ratio between the p and s polarized absorptances is given by:
Ap(O)
{ 1 - Rp(0)}
A~(O)
{1 - R~(0)}
2.3. p and s polarized reflectances from observed spectral intensi~ According to Eq. (6), it is necessary to obtain s and p polarized reflectances in order to calculate the estimated mean square electric field. Since an incident beam with a degree of polarization ~ is utilized, reflectance of the film is a combination of p and s polarized reflectances. The p and s polarized reflectances can be calculated from the reflectance of the film by using the degree of polarization and evanescent fields at the surface of the film, eliminating the need for experimental determination. According to Eqs. (1), (2), and (6) the following expressions
{1 + R p ( 0 ) }
{1 +R,(0)}
(6)
where (E2z(0))k= 0 and (E~t(0))k= 0 are the evanescent field at depth z and the surface of the film, respectively, and dp is the penetration depth. The factor (1 + Rl(0))/2 is introduced as an attenuation factor to compensate for the absorption at that peak position.
(8)
-,)
(Eop(O))k=o (E2,(O))k=oe-2z/d,
(7)
_/311 + R p ( 0 ) } + R~(0)}
(9)
where [3 is the ratio between the p and s polarized evanescent fields at the surface of the film. According to Eqs. (4) and (9), the following quadratic equations for the p and s polarized reflectances can be obtained. ( / 3 - 1)(1 +fi)n2p(0) + 2 { ( / 3 + 1)
-(/3-
1)Rp( O)}Rp( O) + ( / 3 - 1)(1 - f i )
- 2(/3 + I)R~(0)
= 0,
(/3-1)(1-fi)R~(0)-2{(/3+
(10)
1)
+(/3- 1)l%( O)}Rd O) + (/3- 1)(l + + 2( /3 + 1)Rp(0) = O.
(11)
Reflectances of the film according to s and p polarized incident beams are then given by:
-b+__ V ~ ' - 4 a c R,(O) =
2a
(12)
4
S. Ekgasit, H. Ishida/Vibrational Spectroscopy 13 (1996) 1-9
For p polarization, a=(fl-
1)(1 + ~ ) ,
(13)
b = 2{( f l + 1) - ( f l - 1)R~(0)},
(14)
c=(fl-1)(1-~)-2(fl+l)R~(O).
(15)
2.4. Depth profiling by multiple angle ATR FTIR spectroscopy
For s polarization, a = ( f l - 1)(1 - ~ ) ,
(16)
b=-2{(fl+l)+(fl-1)R~(O)},
(17)
c=(fl-1)(l+~)+2(fl+l)Rp(O)
(18)
where the corrected s or p polarized reflectance is the root of Eq. (12) that satisfied the condition 0 <_Rt(O) _< 1. The comparison between reflectances calculated by exact optical theory and those from the above method at various extinction coefficients are shown in Fig. 1. The differences in results from the two methods are insignificant at small extinction coefficients. At high extinction coefficients and small incident angles, there are small deviations of the -":- 1
.
reflectances calculated by the above method from those by exact optical theory due to the increased significance of the difference between the mean square electric field and the evanescent field.
0
~
It is well known that ATR FTIR spectroscopy is a promising technique for the investigation of depth dependent properties of a polymer film [2]. Spectral intensity of the film is a function of both the film characteristics and incident beam parameters. By observing the spectral intensities of the probing species at various incident angles, the depth dependent properties of that species can be obtained. The probing frequency should be one of the characteristic frequencies of that species and well defined. By substituting Eq. (2) into Eq. (5), absorptance of the film at frequency v with an incident beam of degree of polarization ff and angle 0 is given by:
Ap(O) = - -
n o cos 0 j=t [
njkj
(1 +~)(EZzp(O))
zj
o• 0.9 '~
+ ( 1 -~)(E2,(Ol)}dz].
0.8 I
I
I
I
Since depth profiling by multiple angle ATR FTIR spectroscopy utilizes absorptance at various angles of incidence, absorptance of the film at those angles are given by:
~. 1.0 ~ ~ o
0.8
~
O~
-
"~
~
~
0.6 V I
k=O.2
,
,
I
I
t.O 0.8
30
~
2 7rv
n
Rp
~--~----
A~(02) k = 0.4
I
40
I
I
I
50 60 70 angleof incidence(degree)
2 njk) fzJ+,(Ez(O,))~dz,
n o cos 01 j= ~
0.6 "~ 0.4 /
Ap(O,) = - 2~rv -
~-n,
(19)
80
Fig. 1. Comparisons between p and s polarized reflectances calculated fi'om exact optical theory (dotted lines) and those from Eq. (12) (solid lines) at various extinction coefficients. The film is assumed to be infinitely thick. The optical parameters are n o = 4.0, nfilm = 1.5, v = 1000 cm- l, and b = 0. Reflectance of the film as a function of incident angle is shown by the broken line.
zj
Zj+ ]
2
E njkjfzj
2"rrv n f zj+ 1 Ap(03)= n0cos03 E n j k j L j= 1 zj
(E2(O3))Pdz (20)
2~
n
I" Z j + l _
Z njkjJzj
A p( Om) n°c°s Omj = l
2
(Ez(Om))pdz
where (EZz(0i))~ = (1 + ~)(E2p(01)) + (1 ~b)(E2s(0i)). The above angle dependent absorp-
S. Ekgasit, H. Ishida / Vibrational Spectroscopy 13 (1996) 1-9
tances can be simplified into a simple matrix form as: (21)
A=E.X
where
E =
-el 1
el2
el3
. ..
eln
e21
e22
e23
•..
eln
e31
e32
e33
•..
eln
eml
eml
eml
...
emn
(22)
2Try fzj+~( % = noCO---ssOi zj E2(Oi))pdz' (i= 1,2,3 ..... m;j=1,2,3
. . . . . n),
(23)
-r/lk 1
X =
n2 k2 n3k3 ,
(24)
nnkn
A=
-Ap( O,) aA02) A (03)
(25)
A~(Om) Matrix E is the electric field matrix. Element eij of matrix E represents an integration of the electric field in the j-th layer multiplied by an angle dependent parameter (2'rrv)/(n o cos 0i). Matrix A is the matrix of absorptance. The i-th element of the matrix represents absorptance of the film at the angle of incidence 0 i. Matrix X represents the complex refractive index profile of the film. The j-th element of the matrix is the product of the refractive index and the extinction coefficient at the j-th stratified layer. In order to obtain matrix X, the above linear equations are solved consecutively. The method for calculating the refractive index and extinction coefficient of each stratified layer from matrix X is given elsewhere [1].
5
3. Results and discussion
The new depth profiling method consists of three calculation steps including the estimation of the complex refractive index profile, linear least square fitting of absorptances, and nonlinear fitting of reflectances. Since a stratified medium approach is applied to the film, the appropriate stratified medium should have the same spectral intensities as those of the film. In our calculation, the stratified medium is defined based on the decay characteristic of the evanescent field at the smallest incident angle. The thickness of the stratified medium is given by the working depth at that angle since molecular characteristics of the film beyond the working depth have insignificant contribution to the spectral intensities. The thickness of each layer in the stratified medium is defined by the decay characteristic of the evanescent field. The method for defining the stratified medium from a set of angle dependent refiectances is given elsewhere [ 1]. Since the number of data point (i.e., number of angle dependent reflectances) is determined by the number of incident angles in the experiment, which is less than 20 depending on the angle resolution of the variable angle attachment. In order to obtain the estimated complex refractive index profile with high accuracy and apply singular value decomposition (SVD) to matrix E, angle dependent spectral intensities are interpolated to the same number as that of the stratified layers (i.e., m --- n) by polynomial interpolation. The detail on solving linear equation by SVD method is given elsewhere [1,18]. The step-by-step depth profiling calculation by this method is shown in Fig. 2. In the first step, the estimated complex refractive index profile is calculated by solving linear equations of absorptance (i.e., Eq. (21)). The SVD method is used to solve the equations. Matrix A is constructed from the interpolated reflectances. Since the complex refractive index profile of the film is not known, the estimated mean square electric fields are used to constructed matrix E instead of the exact mean square electric fields. The estimated mean square electric fields are calculated from evanescent field and the interpolated reflectances (i,e., Eq. (6)). The p and s polarized reflectances are calculated from the interpolated reflectance using methods given in Eqs. (7)-(18). It
6
S. Ekgasit, H. lshida/Vibrational Spectroscopy 13 (1996) 1-9
is defined as the reconstructed complex refractive
defineprofilingfrequencyv) V ( definestratifiedmedium) x7 calculateRp andRs fromvinterpolatedreflectance I
index profile of the film. l
generatematrixE fromthe estimatedmeansauare electricfields 1 ( generatematrixA frominterpolatedreflectance) solvelinearequationsof absorptanceby SVD method 1
~
[
~
the.._..._~estimatedcomplexreZractiveindexprofile_._~._~ LINEARLEASTSQUAREFITTING OF ABSORPTANCE
I
=~
[ NONLINEARFITTINGOF REFLECTANCE I
-
complex refractive index profile of the film ~
~
Fig. 2. A flow chart for depth profiling by multiple angle ATR FTIR spectroscopy using an incident beam with a known degree of polarization.
should be noted that the estimated mean square electric fields are used only in the first step. The estimated complex refractive index profile is then used as a trial profile for linear least square fitting of absorptance in the second step. The mean square electric fields corresponding to the estimated profile are calculated and used to construct matrix E. Matrix A, in this step, is also constructed from the interpolated reflectances. The method for the mean square electric field calculation is given elsewhere [1,10]. In the course of linear least square fitting, iterative calculation is used in order to obtain a better complex refractive index profile (i.e., matrix E for a current calculation is constructed from the complex refractive index profile obtained from the previous calculation). The converged profile from the linear least square fitting is defined as the complex refractive index profile from linear fitting. The profile is then used as a trial profile for nonlinear fitting between reflectances from experiment and those from exact optical theory. The Levenberg-Marquardt method is used as an algorithm for the nonlinear fitting [1,18]. The converged profile from the fitting
In order to show the applicability of this approach, a set of angle dependent spectra of a known profile are calculated by spectral simulation using matrix method. A white noise spectrum with a constant amplitude is then added to those spectra. Spectral intensities of the noise-added spectra at frequency v are then used as experimental data. The comparison between the known profile and reconstructed profile is made. Fig. 3b shows a set of fifteen angle dependent reflectances at 1000 cm -l and their corresponding interpolated reflectances. The reflectances are calculated from the complex refractive index profile shown in Fig. 3a with the added white noise of amplitude 0.0001 a.u. Although both refractive index and extinction coefficient are depth dependent parameters, a constant refractive index profile (i.e., n; = 1.5) is used for all depth profiling calculations. Fig. 4 shows the corresponding extinction coefficient profiles of the reflectances shown in Fig. 3b, which are obtained at various steps of the
,,ooI . o.1 8 o 1.497 0.0
a
0.0
,
0.5
1.0 1.5 depth (micron)
2.0
2.5
i i i 50 60 70 angle of incidence (degree)
80
g
5"
1.00 = 0.95
8 t~
o.9o 0.85
"~ 0.80 0.751 30
i 40
Fig. 3. An example of data used for depth profiling calculation. (a) The complex refractive index profile of the film. (b) A set of fifteen angle dependent reflectances with added white noise (filled squares) that are used as experimental data. The corresponding interpolated reflectances of the experimental data (open circles) are used for depth profiling calculation in steps 1 and 2. The optical parameters are n o = 4.0, nfitm = 1.5, v = 1000 c m - 1 and = 0. The non-absorbing substrate has a refractive index n~ = 1.5.
S. Ekgasit, H. Ishida / Vibrational Spectroscopy 13 (1996) 1-9 0.20
"
~
,
0.4
"'" ......
'o,
0.15
oo
'
--
the known profile estimated profile - linear least square fitting nonlinear fitting
7
"%
g 0.3
0.10
~_
g
'%
0.2
"'",,
0.05 '~ 0.1 0.00
-0.05 0
I
I
1
2
0.0 0.0
I
J
0.5
depth (micron) Fig. 4. The reconstructed extinction coefficient profiles of the film at various steps of depth profiling calculation. The profiles are calculated from reflectances shown in Fig. 3b. The optical parameters are the same as those of Fig. 3.
depth profiling calculation. The figure indicates the convergence of the reconstructed profiles towards the known profile. The converged extinction coefficient profile from nonlinear fitting is assigned as the extinction coefficient profile of the film. The reconstructed extinction coefficient profiles corresponding to incident beams with various degrees of polarization are shown in Fig. 5. The reconstructed extinction coefficients are almost the same and very close to the known profile. The profile with s polarized
i:i
0.20
theknown pro,ie 5 =-1.0
E 0.15 "5
I
1.0 1.5 depth (micron)
2.0
2.5
Fig. 6. The reconstructed linear decreasing extinction coefficient profiles of different magnitudes of extinction coefficient. The solid lines indicate the reconstructed profiles. The dotted lines indicate the profiles that used to calculate angle dependent spectra. The optical parameters and number of stratified layers are the same as those in Fig. 5.
incident beams ( } = - 1) seem to have a slight deviation at the end of the profile compared to those with p polarized incident beams ( ~ = + 1). The reconstructed profiles from various types of complex refractive index profiles are shown in Figs. 6-9. The linear decreasing profiles with different extinction coefficient magnitudes are shown in Fig. 6. The linear decreasing profiles with different pro-
0.20
0.15
`5= 0.0 `5 = +1.0
o
o~ 0.10
o 0.10
"-%.
0.05
0.05
0.00
0.00
I
0.0
0.5
I
I
1.0 1,5 depth (micron)
I
I
2.0
2.5
Fig. 5. The comparison between the reconstructed extinction coefficient profiles at various degrees of polarization with the profiles that used to calculate angle dependent spectra (the known profiles). The optical parameters are n o = 4.0, nfilm = 1.5, n s = 1.5, and v = 1000 c m - i. Number of stratified layers is 200.
0.0
0.5
I
1.0
I
I
1.5 2.0 depth (micron)
[
2.5
"'"% l 3.0
3.5
Fig. 7. The reconstructed linear decreasing extinction coefficient profiles of different profile regions. The solid lines indicate the reconstructed profiles. The dotted lines indicate the profiles that are used to calculate angle dependent spectra. The optical parameters and number of stratified layers are the same as those in Fig. 5.
8
S. Ekgasit, H. lshida / Vibrational Spectroscopy 13 (1996) 1-9 0.30 0.2 . ~ _ . . . . . . . _ . .
the known profile
c 0~
noise
= lx10 -s a.u.
noise
= lx10 "~ a.u.
noise = lx10 -3 a.u.
"5 0.20
._~ 0.1
.~_ 0.10
-~ 0.00
i
0.0
0.5
i
i
i
1.0
1.5
2.0
~-.Z, 0.0 0.0
2.5
I
0.5
depth (micron)
i
I
i
1.0
1.5
2.0
.... 2.5
depth (micron)
Fig. 8. The reconstructed linear decreasing extinction coefficient profiles of different substrate. The solid lines indicate the reconstructed profiles. The dotted lines indicate the profiles used to calculate the angle dependent spectra. The optical parameters and number of stratified layers are the same as those in Fig. 5.
Fig. 10. The reconstructed extinction coefficient profiles calculated from simulated angle dependent spectra with different added noise amplitudes. The optical parameters and number of stratified layers are the same as those in Fig. 5.
indicates that the higher the noise amplitude, the lower the accuracy of the reconstructed profile. file regions are shown in Fig. 7. The exponential decay profiles with various exponential factors are shown in Fig. 8. The reconstructed profiles with absorbing substrates are shown in Fig. 9. The effect of noise amplitude on the accuracy of the reconstructed profiles is shown in Fig. 10. The figure
0.3
.o_ 0.2 0)
8 0.1
0.0 0
I
I
1
2
4. Conclusions
A depth profiling analysis via multiple angle ATR FTIR spectroscopy using an incident beam with arbitrary degree of polarization has been developed. The analysis does not require any estimated profile or prior knowledge of the complex refractive index profile of the film, however the degree of polarization of the incident beam has to be known. The applicability of this approach has been theoretically demonstrated. This method has been successfully applied to the simulated ATR spectra with various noise levels, degree of polarization, complex refractive index profiles, and substrates. The analysis has shown the high accuracy of the reconstructed profiles compared to the known profiles.
References 3
depth (micron) Fig. 9. The reconstructed exponential decay extinction coefficient profiles of different decay functions. The solid lines indicate the reconstructed profiles. The dotted lines indicate the profiles that used to calculate angle dependent spectra. The optical parameters and number of stratified layers are the same as those in Fig. 5 with v = 800 cm 1.
[1] [2] [3] [4] [5]
S. Ekgasit and H. Ishida, Appl. Spectrosc., in press. H. Ishida, Bull. Inst. Chem. Res. Kyoto Univ. 71 (1993) 190. K. Ohta and R. Iwamoto, Appl. Opt. 29 (1990) 1952. L.J. Fina and G. Chen, Vib. Spectrosc. 1 (1991) 353. R.A. Shick, J.L. Koenig and H. Ishida, Appl. Spectrosc. 47 (1993) 1237. [6] L.J. Fina, Appl. Spectrosc. Rev. 29 (1994) 309.
S. Ekgasit, H. lshida / Vibrational Spectroscopy 13 ~1996) 1-9
[7] M. Yanagimachi, M. Toriumi and H. Masuhara, Appl. Spectrosc. 46 (1992) 832. [8] J. Huang and M.W. Urban, Appl. Spectrosc. 47 (1993) 973. [9] M. Born and E. Wolf, Principle of Optics, 6th Ed. (Pergamon Press, Oxford, 1980). [10] W.N. Hansen, J. Opt. Soc. Am. 58 (1968) 380. [1 I] K. Ohta and H. Ishida, Appl. Spectrosc. 30 (1985)418. [12] F.M. Mirabella, Jr., Appl. Spectrosc. Rev. 21 (1985) 45. [13] R.W. Hunter, J. Opt. Soc. Am. 55 (1965) 1197. [14] D.S. Kliger, J.W. Lewis and C.E. Randall, Polarized Light in Optics and Spectroscopy (Academic Press, San Diego, 1990).
9
[15] W.N. Hansen, Spectrochim. Acta 21 (1965) 815. [16] J.D.E. Mclntyre, in: Advances in Electrochemistry and Electrochemical Engineering, Vol. 9, ed. R.H. Muller (Wiley, New York, 1973). [17] W.N. Hansen, in: Advances in Electrochemistry and Electrochemical Engineering, Vol. 9, ed. R.H. Muller (Wiley, New York, 1973). [18] W.H. Press, S.A. Teukolsky, W.T. Vetterting and B.P. Flannery, Numerical Receipes in FORTRAN: The Art of Scientific Computing, 2nd Ed. (Cambridge University Press, New York, 1992).
Vibrational Spectroscopy 13 (1996) 1-9
Optical depth profiling by attenuated total reflection Fourier transform infrared spectroscopy using an incident beam with arbitrary degree of polarization Sanong Ekgasit, Hatsuo Ishida * Department of Macromolecular Science, Case Western Reserve University, Cleveland, OH 44106-7202. USA
Received 16 January 1996; accepted 6 April 1996
Abstract A depth profiling analysis via multiple angle attenuated total reflection Fourier transform infrared spectroscopy using an incident beam with arbitrary degree of polarization has been developed. A guessed complex refractive index profile is not required for the analysis. The analysis consists of three steps: estimation of the complex refractive index profile by the estimated mean square electric field, linear least square fitting of absorptances, and nonlinear fitting of reflectances. The applicability of this approach with various types of profile and degrees of polarization has been shown by simulated spectra with added white noise. The effect of amplitudes of noise on the accuracy of this approach has been investigated. Keywords." ATR FTIR; Multiple angle ATR FrlR spectroscopy;Depth profiling; Linear least square fitting; Nonlinear fitting
1. Introduction One of the unique characteristics of ATR FFIR spectroscopy is that it provides depth dependent information as a function of refractive index of the incident medium, complex refractive index of the film, angle, and frequency of the incident beam [1,2]. The depth dependent properties such as concentration, conformation, and degree of polymerization of a film can, then, be obtained from multiple angle ATR spectra. There are two depth profiling techniques that take advantage of this unique characteristic, namely Laplace transform and exact optical theory techniques [3-8]. Although the applicability of
* Corresponding author. Tel.: + 1-216-3684285; fax: + 1-2163684164.
the latter method is not limited to spectral regions with small absorption as is the former method, both methods require an estimated or guessed complex refractive index profile for their calculations. By using the estimated profile, the experimental data is then fitted to the calculated data. Even though the calculations of both techniques are straightforward, it is difficult for one to estimate the right profile even if some information about the profile is known. Recently, we have developed a new depth profiling method that overcomes the above difficulties [ 1]. This method utilizes multiple angle ATR spectra taken with a p or s polarized incident beam. The analysis consists of two steps: linear least square fitting of the absorptances and nonlinear fitting of the reflectances. The estimated complex refractive index profile is obtained by solving a set of linear
0924-2031/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PIIS0924-2031(96)00028-8
2
S. Ekgasit, H. lshida/Vibrational Spectroscopy 13 (1996) 1-9
equations of absorptance. The profile is then used as a trial profile for nonlinear fitting of reflectances from experiments to those from exact optical theory. The purpose of this paper is to develop a multiple angle ATR depth profiling method for a set of spectra taken with an arbitrarily polarized incident beam. This will allow an infrared beam of a spectrometer without a polarizer to be used as long as the degree of polarization caused by the instrument is known. In order to obtain the complex refractive index profile of the film, the same approach for depth profiling calculation mentioned above with some modifications is utilized.
2. Theory
2.1. Spectral intensity with an arbitrary, polarized incident beam A film with anisotropic properties in the depth direction can be considered as a stratified medium. The stratified medium is defined as a stack of n isotropic thin films. Each film or layer of the stratified medium is considered homogeneous and parallel plane-bound with the neighboring layers. The characteristics of the j-th layer is given by its complex refractive index, ~j = nj + ikj, and thickness, dj. The reflectance, transmittance, and absorptance of the film are the same as those of the stratified medium [9,10]. For an ATR spectrum of a thick film, reflection lost from the spectrum is caused solely by the absorption of the film. The relationship between reflectance, Rt(0), and absorptance, At(0), in ATR FTIR spectroscopy is given by [1,4-6,11]:
A,(O) = 1 - R , ( 0 )
(1)
where l indicates polarization (i.e., s or p) and 0 is the incident angle. The absorptance of the film can be expressed in terms of characteristic parameters of the stratified medium as [1-3,11,12]: j ~ 1njkj f.;'+'(E2zt(O))dz At(O ) - - 4"n'v n o cos 0 .=
(2)
where (Ez2/(0)) is the mean square electric field at depth z, zj is the depth at the interface between the
( j - 1)-th and j-th layers, z,+ 1 is the depth where the mean square electric field is negligibly small. By adopting the working depth concept, molecular characteristics beyond the working depth have insignificant contributions to the spectral intensity [1]. The working depth is defined as the depth at which the evanescent field has decayed to 0.25% of its value the surface. The working depth is approximately three times the penetration depth [12]. The depth Zn+ 1 is then defined by the working depth. It should be noted that the evanescent field is the special name for the mean square electric field in a non-absorbing medium (i.e., k = 0). It can be calculated if the refractive index of the film is known. For an arbitrary polarized incident beam, the degree of polarization of the beam, ,~, is given by [13,14]:
ip-l ip+(,
(3)
where lp and I s are p and s polarized intensities of the incident beam, respectively. The degree of polarization lies in the range - 1 < ~ < 1. The degree of polarization of the p polarized incident beam is equal to + 1 while that of the s polarized beam is equal to - 1 . Natural light is defined as an non-polarized beam since the intensities of p and s polarized components of the beam are the same. As a result, the degree of polarization of natural light is equal to zero. Unlike natural light, the beam of a spectrometer is partially polarized [ 15]. Its degree of polarization must be determined in order to use this method for depth profiling calculation. With an incident beam of degree of polarization ~, the spectral intensity of a film is the combination of the intensities corresponding to the p and s components. In ATR FTIR spectroscopy, reflectance and absorptance of a film corresponding to an incident beam with a degree of polarization ~ (i.e., Rp(0) and A~(0)) are given by the following expressions [15,16]:
R~(O) = ½{(1 + f i ) R p ( 0 ) + (1 - , ~ ) R s ( 0 ) } ,
(4)
A~(O) = 2 { ( l + f i ) a p ( 0 )
(5)
+(1-~)as(0)}
where Rp(e) and Ap(0) are reflectance and absorp-
S. Ekgasit, H. lshida / Vibrational Spectroscopy 13 ¢1996) 1 - 9
tance of the film with a p polarized incident beam while R~(0) and A,(0) are those with an s polarized incident beam, respectively.
are obtained, respectively, for p and s polarized incident beams: Ap(0)= 1-Rp(0) 2~-v{1 + Rp(0)}
2.2. The estimated mean square electric field = The mean square electric field is significantly different from the evanescent field, especially at positions where the extinction coefficient is large [1]. For a peak position with a small extinction coefficient, the attenuation effect by the absorption is small. The mean square electric field can be estimated from the reflectance and evanescent field at the same conditions (i.e., refractive indices of incident medium and film, angle, frequency, and polarization of the incident beam). Under the above condition, the estimated mean square electric field is assumed to have the same decay characteristic as that of the evanescent field and is given by [17]: 1 +R,(O)
___-
2
1 + R,(0) 2
3
/ 2 ,E.(o)L=o
n0cos 0
X L n j k j fr z j + 1e- 2 :/drdz, .j= I .# A~(O) = 1 -R~(O) 27rv{1 + Rs(0)}
=
n o cos 0
X ~ n j k j f zJ+l e --:/d~ "~ dz. j=
1
The ratio between the p and s polarized absorptances is given by:
Ap(O)
{ 1 - Rp(0)}
A~(O)
{1 - R~(0)}
2.3. p and s polarized reflectances from observed spectral intensi~ According to Eq. (6), it is necessary to obtain s and p polarized reflectances in order to calculate the estimated mean square electric field. Since an incident beam with a degree of polarization ~ is utilized, reflectance of the film is a combination of p and s polarized reflectances. The p and s polarized reflectances can be calculated from the reflectance of the film by using the degree of polarization and evanescent fields at the surface of the film, eliminating the need for experimental determination. According to Eqs. (1), (2), and (6) the following expressions
{1 + R p ( 0 ) }
{1 +R,(0)}
(6)
where (E2z(0))k= 0 and (E~t(0))k= 0 are the evanescent field at depth z and the surface of the film, respectively, and dp is the penetration depth. The factor (1 + Rl(0))/2 is introduced as an attenuation factor to compensate for the absorption at that peak position.
(8)
-,)
(Eop(O))k=o (E2,(O))k=oe-2z/d,
(7)
_/311 + R p ( 0 ) } + R~(0)}
(9)
where [3 is the ratio between the p and s polarized evanescent fields at the surface of the film. According to Eqs. (4) and (9), the following quadratic equations for the p and s polarized reflectances can be obtained. ( / 3 - 1)(1 +fi)n2p(0) + 2 { ( / 3 + 1)
-(/3-
1)Rp( O)}Rp( O) + ( / 3 - 1)(1 - f i )
- 2(/3 + I)R~(0)
= 0,
(/3-1)(1-fi)R~(0)-2{(/3+
(10)
1)
+(/3- 1)l%( O)}Rd O) + (/3- 1)(l + + 2( /3 + 1)Rp(0) = O.
(11)
Reflectances of the film according to s and p polarized incident beams are then given by:
-b+__ V ~ ' - 4 a c R,(O) =
2a
(12)
4
S. Ekgasit, H. Ishida/Vibrational Spectroscopy 13 (1996) 1-9
For p polarization, a=(fl-
1)(1 + ~ ) ,
(13)
b = 2{( f l + 1) - ( f l - 1)R~(0)},
(14)
c=(fl-1)(1-~)-2(fl+l)R~(O).
(15)
2.4. Depth profiling by multiple angle ATR FTIR spectroscopy
For s polarization, a = ( f l - 1)(1 - ~ ) ,
(16)
b=-2{(fl+l)+(fl-1)R~(O)},
(17)
c=(fl-1)(l+~)+2(fl+l)Rp(O)
(18)
where the corrected s or p polarized reflectance is the root of Eq. (12) that satisfied the condition 0 <_Rt(O) _< 1. The comparison between reflectances calculated by exact optical theory and those from the above method at various extinction coefficients are shown in Fig. 1. The differences in results from the two methods are insignificant at small extinction coefficients. At high extinction coefficients and small incident angles, there are small deviations of the -":- 1
.
reflectances calculated by the above method from those by exact optical theory due to the increased significance of the difference between the mean square electric field and the evanescent field.
0
~
It is well known that ATR FTIR spectroscopy is a promising technique for the investigation of depth dependent properties of a polymer film [2]. Spectral intensity of the film is a function of both the film characteristics and incident beam parameters. By observing the spectral intensities of the probing species at various incident angles, the depth dependent properties of that species can be obtained. The probing frequency should be one of the characteristic frequencies of that species and well defined. By substituting Eq. (2) into Eq. (5), absorptance of the film at frequency v with an incident beam of degree of polarization ff and angle 0 is given by:
Ap(O) = - -
n o cos 0 j=t [
njkj
(1 +~)(EZzp(O))
zj
o• 0.9 '~
+ ( 1 -~)(E2,(Ol)}dz].
0.8 I
I
I
I
Since depth profiling by multiple angle ATR FTIR spectroscopy utilizes absorptance at various angles of incidence, absorptance of the film at those angles are given by:
~. 1.0 ~ ~ o
0.8
~
O~
-
"~
~
~
0.6 V I
k=O.2
,
,
I
I
t.O 0.8
30
~
2 7rv
n
Rp
~--~----
A~(02) k = 0.4
I
40
I
I
I
50 60 70 angleof incidence(degree)
2 njk) fzJ+,(Ez(O,))~dz,
n o cos 01 j= ~
0.6 "~ 0.4 /
Ap(O,) = - 2~rv -
~-n,
(19)
80
Fig. 1. Comparisons between p and s polarized reflectances calculated fi'om exact optical theory (dotted lines) and those from Eq. (12) (solid lines) at various extinction coefficients. The film is assumed to be infinitely thick. The optical parameters are n o = 4.0, nfilm = 1.5, v = 1000 cm- l, and b = 0. Reflectance of the film as a function of incident angle is shown by the broken line.
zj
Zj+ ]
2
E njkjfzj
2"rrv n f zj+ 1 Ap(03)= n0cos03 E n j k j L j= 1 zj
(E2(O3))Pdz (20)
2~
n
I" Z j + l _
Z njkjJzj
A p( Om) n°c°s Omj = l
2
(Ez(Om))pdz
where (EZz(0i))~ = (1 + ~)(E2p(01)) + (1 ~b)(E2s(0i)). The above angle dependent absorp-
S. Ekgasit, H. Ishida / Vibrational Spectroscopy 13 (1996) 1-9
tances can be simplified into a simple matrix form as: (21)
A=E.X
where
E =
-el 1
el2
el3
. ..
eln
e21
e22
e23
•..
eln
e31
e32
e33
•..
eln
eml
eml
eml
...
emn
(22)
2Try fzj+~( % = noCO---ssOi zj E2(Oi))pdz' (i= 1,2,3 ..... m;j=1,2,3
. . . . . n),
(23)
-r/lk 1
X =
n2 k2 n3k3 ,
(24)
nnkn
A=
-Ap( O,) aA02) A (03)
(25)
A~(Om) Matrix E is the electric field matrix. Element eij of matrix E represents an integration of the electric field in the j-th layer multiplied by an angle dependent parameter (2'rrv)/(n o cos 0i). Matrix A is the matrix of absorptance. The i-th element of the matrix represents absorptance of the film at the angle of incidence 0 i. Matrix X represents the complex refractive index profile of the film. The j-th element of the matrix is the product of the refractive index and the extinction coefficient at the j-th stratified layer. In order to obtain matrix X, the above linear equations are solved consecutively. The method for calculating the refractive index and extinction coefficient of each stratified layer from matrix X is given elsewhere [1].
5
3. Results and discussion
The new depth profiling method consists of three calculation steps including the estimation of the complex refractive index profile, linear least square fitting of absorptances, and nonlinear fitting of reflectances. Since a stratified medium approach is applied to the film, the appropriate stratified medium should have the same spectral intensities as those of the film. In our calculation, the stratified medium is defined based on the decay characteristic of the evanescent field at the smallest incident angle. The thickness of the stratified medium is given by the working depth at that angle since molecular characteristics of the film beyond the working depth have insignificant contribution to the spectral intensities. The thickness of each layer in the stratified medium is defined by the decay characteristic of the evanescent field. The method for defining the stratified medium from a set of angle dependent refiectances is given elsewhere [ 1]. Since the number of data point (i.e., number of angle dependent reflectances) is determined by the number of incident angles in the experiment, which is less than 20 depending on the angle resolution of the variable angle attachment. In order to obtain the estimated complex refractive index profile with high accuracy and apply singular value decomposition (SVD) to matrix E, angle dependent spectral intensities are interpolated to the same number as that of the stratified layers (i.e., m --- n) by polynomial interpolation. The detail on solving linear equation by SVD method is given elsewhere [1,18]. The step-by-step depth profiling calculation by this method is shown in Fig. 2. In the first step, the estimated complex refractive index profile is calculated by solving linear equations of absorptance (i.e., Eq. (21)). The SVD method is used to solve the equations. Matrix A is constructed from the interpolated reflectances. Since the complex refractive index profile of the film is not known, the estimated mean square electric fields are used to constructed matrix E instead of the exact mean square electric fields. The estimated mean square electric fields are calculated from evanescent field and the interpolated reflectances (i,e., Eq. (6)). The p and s polarized reflectances are calculated from the interpolated reflectance using methods given in Eqs. (7)-(18). It
6
S. Ekgasit, H. lshida/Vibrational Spectroscopy 13 (1996) 1-9
is defined as the reconstructed complex refractive
defineprofilingfrequencyv) V ( definestratifiedmedium) x7 calculateRp andRs fromvinterpolatedreflectance I
index profile of the film. l
generatematrixE fromthe estimatedmeansauare electricfields 1 ( generatematrixA frominterpolatedreflectance) solvelinearequationsof absorptanceby SVD method 1
~
[
~
the.._..._~estimatedcomplexreZractiveindexprofile_._~._~ LINEARLEASTSQUAREFITTING OF ABSORPTANCE
I
=~
[ NONLINEARFITTINGOF REFLECTANCE I
-
complex refractive index profile of the film ~
~
Fig. 2. A flow chart for depth profiling by multiple angle ATR FTIR spectroscopy using an incident beam with a known degree of polarization.
should be noted that the estimated mean square electric fields are used only in the first step. The estimated complex refractive index profile is then used as a trial profile for linear least square fitting of absorptance in the second step. The mean square electric fields corresponding to the estimated profile are calculated and used to construct matrix E. Matrix A, in this step, is also constructed from the interpolated reflectances. The method for the mean square electric field calculation is given elsewhere [1,10]. In the course of linear least square fitting, iterative calculation is used in order to obtain a better complex refractive index profile (i.e., matrix E for a current calculation is constructed from the complex refractive index profile obtained from the previous calculation). The converged profile from the linear least square fitting is defined as the complex refractive index profile from linear fitting. The profile is then used as a trial profile for nonlinear fitting between reflectances from experiment and those from exact optical theory. The Levenberg-Marquardt method is used as an algorithm for the nonlinear fitting [1,18]. The converged profile from the fitting
In order to show the applicability of this approach, a set of angle dependent spectra of a known profile are calculated by spectral simulation using matrix method. A white noise spectrum with a constant amplitude is then added to those spectra. Spectral intensities of the noise-added spectra at frequency v are then used as experimental data. The comparison between the known profile and reconstructed profile is made. Fig. 3b shows a set of fifteen angle dependent reflectances at 1000 cm -l and their corresponding interpolated reflectances. The reflectances are calculated from the complex refractive index profile shown in Fig. 3a with the added white noise of amplitude 0.0001 a.u. Although both refractive index and extinction coefficient are depth dependent parameters, a constant refractive index profile (i.e., n; = 1.5) is used for all depth profiling calculations. Fig. 4 shows the corresponding extinction coefficient profiles of the reflectances shown in Fig. 3b, which are obtained at various steps of the
,,ooI . o.1 8 o 1.497 0.0
a
0.0
,
0.5
1.0 1.5 depth (micron)
2.0
2.5
i i i 50 60 70 angle of incidence (degree)
80
g
5"
1.00 = 0.95
8 t~
o.9o 0.85
"~ 0.80 0.751 30
i 40
Fig. 3. An example of data used for depth profiling calculation. (a) The complex refractive index profile of the film. (b) A set of fifteen angle dependent reflectances with added white noise (filled squares) that are used as experimental data. The corresponding interpolated reflectances of the experimental data (open circles) are used for depth profiling calculation in steps 1 and 2. The optical parameters are n o = 4.0, nfitm = 1.5, v = 1000 c m - 1 and = 0. The non-absorbing substrate has a refractive index n~ = 1.5.
S. Ekgasit, H. Ishida / Vibrational Spectroscopy 13 (1996) 1-9 0.20
"
~
,
0.4
"'" ......
'o,
0.15
oo
'
--
the known profile estimated profile - linear least square fitting nonlinear fitting
7
"%
g 0.3
0.10
~_
g
'%
0.2
"'",,
0.05 '~ 0.1 0.00
-0.05 0
I
I
1
2
0.0 0.0
I
J
0.5
depth (micron) Fig. 4. The reconstructed extinction coefficient profiles of the film at various steps of depth profiling calculation. The profiles are calculated from reflectances shown in Fig. 3b. The optical parameters are the same as those of Fig. 3.
depth profiling calculation. The figure indicates the convergence of the reconstructed profiles towards the known profile. The converged extinction coefficient profile from nonlinear fitting is assigned as the extinction coefficient profile of the film. The reconstructed extinction coefficient profiles corresponding to incident beams with various degrees of polarization are shown in Fig. 5. The reconstructed extinction coefficients are almost the same and very close to the known profile. The profile with s polarized
i:i
0.20
theknown pro,ie 5 =-1.0
E 0.15 "5
I
1.0 1.5 depth (micron)
2.0
2.5
Fig. 6. The reconstructed linear decreasing extinction coefficient profiles of different magnitudes of extinction coefficient. The solid lines indicate the reconstructed profiles. The dotted lines indicate the profiles that used to calculate angle dependent spectra. The optical parameters and number of stratified layers are the same as those in Fig. 5.
incident beams ( } = - 1) seem to have a slight deviation at the end of the profile compared to those with p polarized incident beams ( ~ = + 1). The reconstructed profiles from various types of complex refractive index profiles are shown in Figs. 6-9. The linear decreasing profiles with different extinction coefficient magnitudes are shown in Fig. 6. The linear decreasing profiles with different pro-
0.20
0.15
`5= 0.0 `5 = +1.0
o
o~ 0.10
o 0.10
"-%.
0.05
0.05
0.00
0.00
I
0.0
0.5
I
I
1.0 1,5 depth (micron)
I
I
2.0
2.5
Fig. 5. The comparison between the reconstructed extinction coefficient profiles at various degrees of polarization with the profiles that used to calculate angle dependent spectra (the known profiles). The optical parameters are n o = 4.0, nfilm = 1.5, n s = 1.5, and v = 1000 c m - i. Number of stratified layers is 200.
0.0
0.5
I
1.0
I
I
1.5 2.0 depth (micron)
[
2.5
"'"% l 3.0
3.5
Fig. 7. The reconstructed linear decreasing extinction coefficient profiles of different profile regions. The solid lines indicate the reconstructed profiles. The dotted lines indicate the profiles that are used to calculate angle dependent spectra. The optical parameters and number of stratified layers are the same as those in Fig. 5.
8
S. Ekgasit, H. lshida / Vibrational Spectroscopy 13 (1996) 1-9 0.30 0.2 . ~ _ . . . . . . . _ . .
the known profile
c 0~
noise
= lx10 -s a.u.
noise
= lx10 "~ a.u.
noise = lx10 -3 a.u.
"5 0.20
._~ 0.1
.~_ 0.10
-~ 0.00
i
0.0
0.5
i
i
i
1.0
1.5
2.0
~-.Z, 0.0 0.0
2.5
I
0.5
depth (micron)
i
I
i
1.0
1.5
2.0
.... 2.5
depth (micron)
Fig. 8. The reconstructed linear decreasing extinction coefficient profiles of different substrate. The solid lines indicate the reconstructed profiles. The dotted lines indicate the profiles used to calculate the angle dependent spectra. The optical parameters and number of stratified layers are the same as those in Fig. 5.
Fig. 10. The reconstructed extinction coefficient profiles calculated from simulated angle dependent spectra with different added noise amplitudes. The optical parameters and number of stratified layers are the same as those in Fig. 5.
indicates that the higher the noise amplitude, the lower the accuracy of the reconstructed profile. file regions are shown in Fig. 7. The exponential decay profiles with various exponential factors are shown in Fig. 8. The reconstructed profiles with absorbing substrates are shown in Fig. 9. The effect of noise amplitude on the accuracy of the reconstructed profiles is shown in Fig. 10. The figure
0.3
.o_ 0.2 0)
8 0.1
0.0 0
I
I
1
2
4. Conclusions
A depth profiling analysis via multiple angle ATR FTIR spectroscopy using an incident beam with arbitrary degree of polarization has been developed. The analysis does not require any estimated profile or prior knowledge of the complex refractive index profile of the film, however the degree of polarization of the incident beam has to be known. The applicability of this approach has been theoretically demonstrated. This method has been successfully applied to the simulated ATR spectra with various noise levels, degree of polarization, complex refractive index profiles, and substrates. The analysis has shown the high accuracy of the reconstructed profiles compared to the known profiles.
References 3
depth (micron) Fig. 9. The reconstructed exponential decay extinction coefficient profiles of different decay functions. The solid lines indicate the reconstructed profiles. The dotted lines indicate the profiles that used to calculate angle dependent spectra. The optical parameters and number of stratified layers are the same as those in Fig. 5 with v = 800 cm 1.
[1] [2] [3] [4] [5]
S. Ekgasit and H. Ishida, Appl. Spectrosc., in press. H. Ishida, Bull. Inst. Chem. Res. Kyoto Univ. 71 (1993) 190. K. Ohta and R. Iwamoto, Appl. Opt. 29 (1990) 1952. L.J. Fina and G. Chen, Vib. Spectrosc. 1 (1991) 353. R.A. Shick, J.L. Koenig and H. Ishida, Appl. Spectrosc. 47 (1993) 1237. [6] L.J. Fina, Appl. Spectrosc. Rev. 29 (1994) 309.
S. Ekgasit, H. lshida / Vibrational Spectroscopy 13 ~1996) 1-9
[7] M. Yanagimachi, M. Toriumi and H. Masuhara, Appl. Spectrosc. 46 (1992) 832. [8] J. Huang and M.W. Urban, Appl. Spectrosc. 47 (1993) 973. [9] M. Born and E. Wolf, Principle of Optics, 6th Ed. (Pergamon Press, Oxford, 1980). [10] W.N. Hansen, J. Opt. Soc. Am. 58 (1968) 380. [1 I] K. Ohta and H. Ishida, Appl. Spectrosc. 30 (1985)418. [12] F.M. Mirabella, Jr., Appl. Spectrosc. Rev. 21 (1985) 45. [13] R.W. Hunter, J. Opt. Soc. Am. 55 (1965) 1197. [14] D.S. Kliger, J.W. Lewis and C.E. Randall, Polarized Light in Optics and Spectroscopy (Academic Press, San Diego, 1990).
9
[15] W.N. Hansen, Spectrochim. Acta 21 (1965) 815. [16] J.D.E. Mclntyre, in: Advances in Electrochemistry and Electrochemical Engineering, Vol. 9, ed. R.H. Muller (Wiley, New York, 1973). [17] W.N. Hansen, in: Advances in Electrochemistry and Electrochemical Engineering, Vol. 9, ed. R.H. Muller (Wiley, New York, 1973). [18] W.H. Press, S.A. Teukolsky, W.T. Vetterting and B.P. Flannery, Numerical Receipes in FORTRAN: The Art of Scientific Computing, 2nd Ed. (Cambridge University Press, New York, 1992).